A) \[f(x)\] is continuous at x= 1, 10, 15
B) \[f(x)\] is continuous at \[x=n/3,\] where n is any integer
C) \[\int\limits_{0}^{2/3}{f(x)dx=1/3}\]
D) \[\underset{x\to 2/3}{\mathop{\lim }}\,f(x)=2\]
Correct Answer: C
Solution :
| \[\int\limits_{0}^{2/3}{f(x)dx}\] |
| \[\int\limits_{0}^{2/3}{\left( [x]+\left[ x+\frac{1}{3} \right]+\left[ x+\frac{2}{3} \right] \right)}\,\,dx\] |
| \[=\int\limits_{0}^{1/3}{0dx}+\int\limits_{1/3}^{2/3}{dx}=\frac{2}{3}-\frac{1}{3}=\frac{1}{3}\] |
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